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Dynamical distortions of structural signatures in
molecular high-order harmonic spectroscopy
Marie Labeye, François Risoud, Camille Lévêque, Jérémie Caillat, Alfred
Maquet, Tahir Shaaran, Pascal Salières, Richard Taïeb
To cite this version:
Marie Labeye, François Risoud, Camille Lévêque, Jérémie Caillat, Alfred Maquet, et al.. Dynamical
distortions of structural signatures in molecular high-order harmonic spectroscopy. Physical Review A,
American Physical Society, 2019, 99 (1), pp.013412. �10.1103/PhysRevA.99.013412�. �hal-02148208�
Dynamical distortions of structural signatures in molecular high-order harmonic spectroscopy
Marie Labeye,1François Risoud,1Camille Lévêque,1,2,*Jérémie Caillat,1Alfred Maquet,1Tahir Shaaran,3Pascal Salières,3and Richard Taïeb1
1Sorbonne Université, CNRS, Laboratoire de Chimie Physique-Matière et Rayonnement, 75005 Paris, France
2Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany
3LIDYL, CEA, CNRS, Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France
(Received 16 November 2018; published 11 January 2019)
We study the signature of two-center interferences in molecular high-order harmonic spectra, with an emphasis on the spectral phase. With the help of both ab initio computations based on the time-dependent Schrödinger equation and the molecular strong-field approximation (SFA) as developed by Chiril˘a et al. [Phys. Rev. A 73,
023410(2006)] and Faria [Phys. Rev. A 76,043407(2007)], we observe that the phase behavior is radically different for the short and the long trajectory contributions. By means of Taylor expansions of the molecular SFA, we link this effect to the dynamics of the electron in the continuum. More precisely, we find that the value of the electric field at recombination time plays a crucial role in the shape of the destructive interference phase jump.
DOI:10.1103/PhysRevA.99.013412
I. INTRODUCTION
Over the past few decades, high-order harmonic generation (HHG) has led to an increasing interest in the field of strong laser physics with the development of attosecond science [1,2]. Two main classes of experiments have emerged in this context. First, HHG is a robust tool to generate single or trains of attosecond pulses of coherent extreme ultraviolet (XUV) radiation [3–10], which can be used to perform elec-tronic or nuclear pump-probe spectroscopy on the attosecond scale with angstrom resolution [11–16]. Second, HHG can be employed as a self-probing tool to monitor attosecond dynamics in the generating medium [17–23], e.g., to imple-ment tomography of molecular orbitals [24–26]. References [27,28] provide a comprehensive review on the self-probing applications of HHG.
Most of these studies showed that the intensity and the phase of the generated harmonics encode structural and dy-namical information. While the measurement of their inten-sity, i.e., counting the photon number for each wavelength, is rather straightforward, obtaining their phases was more difficult. At present, several experimental techniques based on interferometry have been mastered to access the harmonic phase, such as RABBIT (reconstruction of attosecond beating by interference of two-photon transition) [29,30], streaking [31,32], TSOIN (two-source optical interferometry) [33–35], PROOF (phase retrieval by omega oscillation filtering) [36], MAMMOTH (mixed approaches for the measurement of the total harmonic phases), CHASSEUR (combined harmonic
*Present address: Wolfgang Pauli Institute c/o Faculty of
Mathe-matics, University of Vienna, Oskar-Morgenstern Platz 1, A-1090 Vienna, Austria and Vienna Center for Quantum Science and Tech-nology, Atominstitut, TU Wien, Stadionallee 2, A-1020 Vienna, Austria.
attosecond spectroscopy by two-source euv interferometry and rabbit) [37], or other approaches [38,39]. These measures of the harmonic phase are crucial to study, e.g., the dynam-ics of autoionizing states or Fano resonances [12,37,40–47], interferences between different ionization paths [33,48–50], and two-center interferences [34,51–57].
To picture the phenomenon of HHG, the well-known three-step model is invoked [58,59]: (i) under the influence of a strong laser field, an electron is first pulled out from the atom or molecule, then (ii) scattered in the continuum and driven back to the parent ion, where (iii) it finally may recombine into its initial state, releasing radiatively its accumulated kinetic energy. The emitted photons constitute the harmonic spec-trum. In this model, the electron in the continuum is consid-ered as a classical particle and thus its motion is characterized by trajectories. Within the strong-field approximation (SFA) [60], which is a fully quantum theory, the notion of electron trajectories (or quantum orbits) in the continuum arises when searching saddle-point solutions in the equations leading to the emitted dipole [61–63], thus validating the three-step picture. Each harmonic of the spectrum is mainly emitted by two possible trajectories of the recolliding electron: the short and the long ones. Both lead to the same kinetic energy but with different times at which the electron recollides with the parent ion. The long trajectories correspond to electrons released before and recolliding after the short ones. Within a quantum theory, they interfere and shape the total spectrum with constructive and destructive patterns.
In the case of molecules, one may envision that electrons ionized from one atomic center and recombining to another will form additional classes of trajectories. Therefore, the structure of the system is encoded through additional inter-ferences in the spectrum, which are reminiscent of Young’s two-slit phenomenon in diatomics [64,65].
In a standard SFA saddle-point approach [66–68], the structure of the molecule is usually introduced via the
tran-sition matrix elements in steps 1 and 3, whereas the electron dynamics, i.e., step 2, remains atomic. Within this approach, destructive interferences are observed in the harmonic spec-trum of diatomic molecules that correspond to the zeros of the recombination dipole matrix element. This results in a discontinuous phase jump of exactly±π rad in the theoretical spectra.
However, this behavior does not compare well with exper-imental data [51,53] nor with ab initio computations based on the exact resolution of the time-dependent Schrödinger equation (TDSE) [64,65,69]. The phase jumps at the destruc-tive interferences are found to be far from sharp: they are smoothed, cover several harmonics and span less than π rad.
Ten years ago, a method accounting for the effect of the molecular structure on the continuum electron dynamics in SFA computations, called molecular SFA, was proposed in [70] and closely investigated in [71]. It relies on the separation of the dipole into four terms, each one exhibiting an additional phase incorporated into the electronic action. The saddle-point approximation is carried out for each term and leads to four groups of possible electron trajectories (with their subsets of short and long paths). They correspond to the ionization from one center and the recollision to either the same center or the other one. To our knowledge, the shape of the phase jumps in such calculations has not been studied or addressed except in Ref. [55], where similar computations have been performed in the specific case of aligned polar CO molecules. Interestingly, the phase jumps are smoothed, which we will confirm with our simulations.
Other models based on the computation of SFA with field-distorted orbitals also display smoothed phase jumps [56,57]. These approaches rely on an adiabatic dressing of the ground state by the instantaneous electric field at recombination. This modifies the shape of the corresponding dipole, and thus the shape of the spectrum. An important feature of this approach is that, if the recombination occurs at a time when the electric field is zero, then the recombination dipole is no longer modified by the dressing, and one recovers a sharp minimum in the spectrum.
We propose here a complete examination of the two-center interferences in homonuclear diatomic molecular mod-els, beyond the smoothing of the structural phase jumps. The motivation for this study arose when investigating HHG amplitude and phase for diatomic molecules as a function of the internuclear distance within ab initio computations based on the TDSE. We observed very different behaviors between the phase jumps when discriminating the short and the long trajectories, in apparent relation with the electric field at the recombination. To obtain physical insight with an analytical theory, we compared with the results of molecular SFA [70,71] and found that they reproduce the ab initio observations. Finally, we performed additional developments of the molecular SFA based on Taylor expansions, and they fully explain the shape of the phase jump in relation to the electric field at the recombination time.
This paper gives a full account of our results following our preliminary studies presented in [72]. It is organized as follows. In Sec. II we introduce the models we used. First, we expose the ab initio computations we performed on a one-dimensional (1D) molecular model by solving the TDSE.
Second, we recall the basic equations of atomic SFA and molecular SFA and expose our further developments based on Taylor expansions of the molecular SFA. The ab initio com-putations will serve as the reference to compare with the SFA computations. In Sec.IIIwe display the results of the ab initio and the molecular SFA computations in the case of a model H2. In Sec.IVwe provide explanations of the behavior of the
phase jumps with the examination of the Taylor expansions of the molecular SFA. We finally summarize our work in Sec.V
and we will draw a qualitative parallel between the field-distorted orbitals computations [56,57] and our developments. In this paper, all the equations are expressed in atomic units (a.u.).
II. METHODS A. Ab initio computations
We consider a diatomic molecule subjected to a low-frequency strong laser field with a 1D model within the single-active electron and fixed nuclei approximation [73], by solving numerically the TDSE. The computational efficiency of these simulations allows us to perform an extensive study of the problem. This approach serves as an exact reference illustrating the validity of our revisited molecular SFA and its interpretation presented hereafter.
We use a double-well potential V (x ) as in [74], where the interaction between the electron and each atomic center is modeled with a regularized Coulomb potential:
V(x )= − 1/2 x−R 2 2 + a2 − 1/2 x+R 2 2 + a2 . (1)
Here R is the internuclear distance, a is the regularization parameter, and x denotes the electron coordinate along the laser polarization ex. First, we will examine the two-center interference condition as a function of R, keeping fixed the ionization potential Ip, thus adjusting the regularization
pa-rameter a for each value of R. We have chosen the one of H2: Ip= 0.567 a.u. = 15.43 eV [75]. Second, we will use
this model for the specific case of H2at equilibrium distance,
where R= 1.4 a.u. with a = 1.1339 a.u.. The TDSE is solved for an electron in the ground state interacting with a two-cycle oscillating electric field E(t )= ELf(t ) sin(ωt )ex, where
f(t )=
sin2ω4t if t∈ [0, 4π/ω],
0 elsewhere. (2)
Here EL denotes the amplitude. Note that for such short
pulses, the maximum of the electric field Emax 0.87EL
is substantially smaller than the amplitude EL. To be more
consistent with the SFA computations, we define the peak intensity of our pulse as the square of Emax, giving
approx-imately IL 120cE2L/1.322, with c the speed of light and
0 the vacuum permittivity. We have set the frequency ω to 0.057 a.u., corresponding to a Ti:sapphire laser of 800 nm wavelength. With such a short IR pulse, the generated spec-trum does not exhibit the usual harmonic comb structure: it is continuous through the entire plateau region.
Within this model, the TDSE for the electronic wave function(x, t ) reads i∂ ∂t(x, t ) = −1 2 ∂2 ∂x2 + V (x) + xE(t ) (x, t ). (3) The HHG spectrum is proportional to the Fourier transform of the time-dependent dipoleD(t ), which is computed as the expectation value of the acceleration at each propagation time step [76]. While interest is often focused on the harmonic in-tensity, we focus on its phase as it encodes the dynamic of the system. The dominant contribution to the HHG phase arises from the intrinsic electron dynamics in the continuum (step 2). It is quadratic in energy and evolves rapidly [8,77]. Thus it might hide the small system-specific signatures in which we are interested. However, the dynamics of the electron freed in the continuum is common to all molecules or atoms of the same ionization potential in the same laser conditions. Computations of the HHG spectra from a reference atom with the same ionization potential are therefore used to remove this “universal” quadratic phase by subtracting it from the molecular HHG phase [8]. Our auxiliary atom is modeled by a single regularized Coulomb potential:
V(x )= −√ 1
x2+ b2, (4)
for which we solve the TDSE with the same laser pulse parameters. Here b= 1.2195 a.u. such that the ionization potential is again 0.567 a.u.
As the computed spectra contain contributions from both the short and long trajectories, we can use their different spatial properties to separate them and sort out their distinct signatures in the overall HHG phase. Classical computations show that the excursion of the electron trajectories in the continuum is characterized by a distance x0= EL/ω2 [58].
The short trajectories never go beyond x0 while the long
trajectories always cross this value. Therefore, we first solved the TDSE with a numerical absorber [78] placed at x0, which
eliminates the wave-function components that go further away, corresponding classically to the long trajectories. As a consequence, the surviving part of the wave function is that of the short trajectories, and we can compute the corresponding dipole Ds(t ). The solution of the same TDSE without the absorber at x0leads to the full dipoleD(t ). Thence, to recover
the long trajectory contributionDl(t ), we subtractDs(t ) from D(t ). One should note that this absorber artificially increases the plateau region, leading to a slightly higher cutoff value.
B. Atomic SFA
In the previous section, we explained the need for an atomic reference in order to remove the quadratic harmonic phase and collect the molecular specific two-center signature. This stands also in the case of molecular SFA computations, hence for completeness we recall in this section the basic equations of atomic SFA [60]. In SFA computations, an abrupt start of the laser field does not affect the spectrum, thus we choose a laser field without an envelope:
E(t )= E(t )ex= ELcos(ωt )ex. (5)
The corresponding vector potential A(t ) is defined such that E(t )= −∂A
∂t . (6)
In the length gauge, the dipole reads Dat() = dt t 0 dt d3p drec[p+ A(t )]
× dion(p+ A(t), t) e−iS(p,t,t
,)
. (7)
Here is the harmonic frequency, S is the action, S(p, t, t,) = t t dτ [p+ A(τ )]2 2 + Ip − t, (8) and dion and drec are the projections of, respectively, the
ion-ization and the recombination dipole matrix elements along the laser polarization, given by
dion(k)= k|xE(t)|ϕ0,
(9) drec(k)= ϕ0| − i∇|k · ex,
with|ϕ0 and |k the atomic ground state and continuum wave
functions, respectively.
The term exp[−iS(p, t, t,)] is rapidly oscillating. Thus, the fivefold integration in Eq. (7) can be reduced to the evaluation of the integrand at the stationary solutions of S, for which ∂S ∂p = 0, ∂S ∂t = 0, and ∂S ∂t = 0. (10)
This saddle-point approximation leads to the search of solu-tions of the following equasolu-tions:
t t dτ[p+ A(τ )] = 0, (11a) [p+ A(t )]2 2 + Ip− = 0, (11b) [p+ A(t)]2 2 + Ip = 0. (11c)
We will denote by (pat, tat, tat) the solutions of Eqs. (11a)–
(11c) to emphasize that they are related to atomic trajectories. Note that the stationary momentum can be written as
pat(tat, tat)= −
tat tat A(τ )dτ
tat− tat . (12) We also define Sat= S(pat, tat, tat,).
The total dipole [Eq. (7)] reduces to
Dat() = C(tat, tat)drec(pat+ A(tat))
× dion(pat+ A(tat), tat)e−iSat. (13)
For the sake of conciseness, no distinction is made between the short or the long trajectories in our notations. As a consequence, the sum over the short and long trajectories in the expression of Dat will not be explicitly indicated in the
following. Moreover, and again for clarity, we dropped the dependence of tat and tat (and consequently of pat) on. In
Eq. (13), C is the saddle-point prefactor, which reads [79] C(tat, tat)= 2π + i(tat− tat) 3/2 1 det(Sp)|at , (14)
where Sp is the Hessian matrix of Sp(t, t)= S(pat(tat, tat), t, t): det(Sp)|at= ∂2S p ∂t2 at ∂2S p ∂t2 at − ∂2S p ∂t ∂t at 2 , (15)
with pat(tat, tat) given in Eq. (12). In the prefactor expression
given in Eq. (14), the first factor arises from the evaluation of the integral, Eq. (7), over momentum at pat and accounts for
the spreading of the electronic wave packet in the continuum. The second factor comes from the evaluation of the integral over t and t.
C. Molecular SFA
In this section, we recall the equations of molecular SFA as they have been developed in Refs. [70,71]. We will use the expression of the recombination and ionization dipole matrix elements as derived in Ref. [71]. These equations are used in the next section (Sec. II D) to obtain analytical expressions by means of Taylor expansions, which is the key point of our work.
The electronic ground state of the target molecule is approximated by a linear combination of atomic orbitals (LCAO), restricted to two atomic orbitals φs centered on the nuclei. Therefore, the recombination and ionization dipole matrix elements along ex for a symmetric state (σg) are expressed as [71] drec(p)= R(p) eip·R2 + e−ip· R 2, (16) dion(p, t)= I1(p, t)eip· R 2 + I 2(p, t)e−ip· R 2, (17) where R(p) = φ¯s(p) √ 2[1+ w(R)]p· ex, (18) Iα(p, t)= − E(t) √ 2[1+ w(R)]· i∂ ¯φs ∂p(p)+ (−1) αR 2φ¯s(p) . (19) Here ¯φs is the momentum representation (Fourier transform) of the atomic orbital φs, and w(R) is the overlap between the two orbitals. In our computations, these orbitals are Gaussian functions [60]. The product of the transition dipole matrix elements thus gives
drec[p+ A(t )] × dion(p+ A(t), t)
= 2 α=1 2 β=1
R[p + A(t )] × Iα(p+ A(t), t)e−iαβ(p,t,t
) , (20) −R/2 0 R/2 x at 1 2 at 11 12 21 22 φs(x+R/2) φs(x) φs(x−R/2) V (x)+xE(t)
FIG. 1. One-dimensional sketch of the molecular potential and the four classes of trajectories α→ β, with α, β ∈ {1, 2} in molec-ular SFA. We present the case of aligned molecules along the laser polarization (R= R ex). We pictured the ionization toward
the positive x-coordinate, i.e., with a negative laser field E(t ). We show by a black solid line the resulting potential, the sum of xE(t ) (dotted line) and of the double-well potential V (x ), and the atomic orbitals φscentered on each nucleus used for the computation of the
transition dipole matrix elements.
where αβ(p, t, t)= (−1)α[p+ A(t)]· R 2 −(−1)β [p+ A(t )] ·R 2. (21)
Here α and β label the initial and final atomic center in the electron excursion, respectively. Thus, the total molecular dipole reads D() = 2 α=1 2 β=1 dt t 0 dt dpR[p + A(t )] × Iα(p+ A(t), t)e−iSαβ(p,t,t ,) . (22)
It is the sum of four terms that can be evaluated independently by searching the stationary points of the modified actions:
Sαβ(p, t, t,) = S(p, t, t,) + αβ(p, t, t). (23) This leads to the following saddle-point equations [71]:
t t dτ[p+ A(τ )] + (−1) αR 2 − (−1) βR 2 = 0, (24a) [p+ A(t )]2 2 + Ip− + (−1) β E(t )·R 2 = 0, (24b) [p+ A(t)]2 2 + Ip+ (−1) α E(t)·R 2 = 0, (24c) for which we find four new groups of stationary solutions (pαβ, tαβ, tαβ ). They correspond to four classes of trajectories, represented in Fig.1. They express the ionization from one center and the recombination to either the same center or the other one. Each class contains a short and a long path. Within the saddle-point approximation, the total dipole thus
reduces to D() = 2 α=1 2 β=1 Cαβ(tαβ, tαβ )R[pαβ+ A(tαβ)] × Iα(pαβ+ A(tαβ ), tαβ )e−iSαβ(pαβ ,tαβ,tαβ ,), (25)
with Cαβ(tαβ, tαβ ) the resulting saddle-point prefactors; see Eq. (14).
We will show in Sec. III that the observations on the phase jumps in the ab initio computations are well reproduced within the molecular SFA. This motivated us to further ana-lyze the molecular SFA analytically and get a proper under-standing of these behaviors. In the following, we express the deviations of the molecular solutions regarding the reference atomic ones by means of Taylor expansions.
D. Taylor expansion of the molecular SFA
1. Stationary points
As we explained, in Sec. II A we will calibrate our nu-merical results with the atomic ones in order to remove the quadratic harmonic phase. Moreover, since internuclear distances (a few Å) are small compared to the travel path of the freed electrons in the continuum (several tens of Å), the times for molecular trajectories are expected to remain close to those of an atomic trajectory. These considerations led us to perform Taylor expansions in powers of the internuclear distance R of the molecular SFA equations, Eq. (24). To remain consistent with our TDSE simulations, where we restricted the study to a 1D case, we will consider in the following the specific case of a model molecule, with only one degree of freedom, aligned along the laser polarization, i.e., R= R exand p= p ex. Moreover, the “natural” absence of a perpendicular component of the momentum allows us to derive equations that reveal straightforward physical insights. The solutions of the molecular saddle point equations (24) are expanded as pαβ = pat+ pαβ(1)+ p (2) αβ + O(R3), (26a) tαβ = tat+ tαβ(1)+ t (2) αβ + O(R 3), (26b) tαβ = tat + tαβ(1)+ tαβ(2)+ O(R3), (26c) where pαβ(1), tαβ(1), and tαβ(1) are proportional to R, and p(2)
αβ, t (2)
αβ, and t (2)
αβ are proportional to R2. Inserting this in Eq. (24), we obtain a linear set of equations for the first-order terms: 0= [pat+ A(tat)]tαβ(1)− [pat+ A(tat)]tαβ(1) + (tat− tat)p (1) αβ+ (−1)α− (−1)β 2 R, (27a)
0= −[pat+ A(tat)]E(tat)tαβ(1)+ [pat+ A(tat)]pαβ(1) +(−1)β
2 E(tat)R, (27b)
0= −[pat+ A(tat)]E(tat)tαβ(1)+ [pat+ A(tat)]p (1) αβ +(−1)α 2 E(t at)R. (27c)
Resolving this set of equations (27) gives p(1) αβ = 0, (28a) t(1) αβ = (−1)βR 2[pat+ A(tat)] , (28b) tαβ(1)= (−1)αR 2[pat+ A(tat)] . (28c)
Using Eqs. (11), the last two equations can be rewritten as1
t(1) αβ = (−1)β R/2 2( − Ip) , (29a) tαβ(1)= i (−1)αR/2 2Ip . (29b)
The same procedure can be performed for the second-order terms; see AppendixA.
TablesIandIIdisplay the numerical values of the station-ary momentum pαβand of the ionization and recollision time deviations tαβ− tat and tαβ − tat and compare them with the
values from the developments including either the first-order term only or both the first- and the second-order terms. An excellent agreement is found between the numerically exact results and the development including up to the second order. Yet, we note that the first-order term is in very good agreement with the exact values, with errors limited to a few percent (around 0.02 a.u.). We will therefore restrict our development to this order. As will be shown, the first-order treatment has the valuable advantage of providing a direct and simple physical picture of the underlying timings of ionization and recollision in diatomic molecules.
First, the ionization time is shifted by a pure imaginary delaytαβ(1), as seen in Eq. (29b), which can be interpreted as a correction to the tunneling time with respect to the atomic case. If the electron is ionized from the center α= 1, located at R/2, we find that Im(t1β(1)) < 0, which means that the
time spent “below the barrier” is smaller. This is consistent with a smaller barrier to cross, as seen in Fig.1. The opposite situation is observed for an electron ionized from center α= 2 for which Im(t2β(1)) > 0, where a larger barrier is to be crossed.
Second, the shift for the recollision times with respect to the atomic times,tαβ(1), Eq. (29a), is now real. For an electron recombining with the center located at R/2 again, i.e., β = 1, we find thattα(1)1 <0, which means that the electron
recol-lides earlier compared to the atomic case. Indeed, as pictured in Fig.1, the electron travels a shorter path. On the contrary, an electron recolliding with center β= 2 for which tα(1)2 >0 travels a longer path.
Note that for both time delaystαβ(1)andtαβ(1), their formal expressions, Eqs. (28), can be regarded as the ratio between the additional distance to cross in the molecule,±R/2, and the
1In these equations and in their interpretation below, we used the
sign convention illustrated in Fig.1, where the barrier is lowered for
TABLE I. Numerical values of the stationary momenta pαβ
so-lution of the saddle point Eqs. (24), compared to its development Eq. (26), restricted either to first or second order, for six harmonics (column denoted by the H symbol) within the plateau of the spectrum generated at a laser intensity of 5× 1014W cm−2.
Short trajectories Long trajectories
H Re Im Re Im p11 21 1.8976 0.14767 0.091138 5.0125× 10−4 31 1.6986 0.089790 0.14357 7.7589× 10−4 41 1.5214 0.061121 0.20458 1.0670× 10−3 51 1.3508 0.044192 0.27847 1.3447× 10−3 61 1.1706 0.033242 0.37518 1.4493× 10−3 71 0.93811 0.027673 0.53362 −5.5265 × 10−4 p12 21 1.8976 0.14769 0.091138 5.0125× 10−4 31 1.6986 0.089793 0.14357 7.7590× 10−4 41 1.5214 0.061122 0.20458 1.0670× 10−3 51 1.3508 0.044192 0.27847 1.3447× 10−3 61 1.1706 0.033242 0.37518 1.4493× 10−3 71 0.93811 0.027673 0.53362 −5.5264 × 10−4 p21 21 1.8975 0.14772 0.091138 5.0176× 10−4 31 1.6986 0.089830 0.14357 7.7668× 10−4 41 1.5214 0.061154 0.20458 1.0681× 10−3 51 1.3508 0.044219 0.27847 1.3460× 10−3 61 1.1706 0.033266 0.37518 1.4507× 10−3 71 0.93811 0.027695 0.53362 −5.5317 × 10−4 p22 21 1.8976 0.14774 0.091138 5.0176× 10−4 31 1.6986 0.089833 0.14357 7.7668× 10−4 41 1.5214 0.061155 0.20458 1.0681× 10−3 51 1.3508 0.044220 0.27847 1.3460× 10−3 61 1.1706 0.033266 0.37518 1.4507× 10−3 71 0.93811 0.027695 0.53362 −5.5316 × 10−4 p2nd 21 1.8976 0.14770 0.091138 5.0152× 10−4 31 1.6986 0.089812 0.14357 7.7631× 10−4 41 1.5214 0.061138 0.20458 1.0676× 10−3 51 1.3508 0.044206 0.27847 1.3454× 10−3 61 1.1706 0.033255 0.37518 1.4501× 10−3 71 0.93811 0.027685 0.53362 −5.5280 × 10−4 pat 21 1.8972 0.14788 0.091148 5.0516× 10−4 = p1st 31 1.6982 0.089978 0.14359 7.8195× 10−4 41 1.5210 0.061302 0.20461 1.0753× 10−3 51 1.3504 0.044360 0.27853 1.3549× 10−3 61 1.1702 0.033397 0.37528 1.4596× 10−3 71 0.93754 0.027841 0.53390 -5.6899× 10−4
velocity of the electron given in Eqs. (11b) and (11c), either classically allowed or forbidden.
We now turn to the Taylor expansion of the different terms entering the expression of the total dipole in Eq. (25) in order to find out how these differences of timing affect the phase of HHG.
2. Total dipole
We first expand the modified action Sαβto first order in R: Sαβ(pαβ, tαβ, tαβ ,) = Sat+ αβ(pat, tat, tat) +∂S ∂t attαβ(1)+ ∂S ∂t at tαβ(1)+O(R 2). (30)
The partial derivatives of the action S with respect to t and t cancel at the atomic stationary points (10), so we simply get
Sαβ(pαβ, tαβ, tαβ ,) = Sat+ αβ(pat, tat, tat)+ O(R 2).
(31) Then we expand the ionization and recombination dipole matrix elements [Eqs. (18) and (19)]:
R[pαβ+ A(tαβ)]= Rat[pat+ A(tat)] +(−1)βR(1)(p at+ A(tat), tat) R +O(R2), (32) where Rat(p)= p ¯φs(p) 2 , (33) R(1)(p, t )= −E(t ) 4 ¯ φs(p) p + ∂ ¯φs ∂p (p) , (34) and Iα(pαβ+ A(tαβ ), tαβ )= Iat(pat+ A(tat), tat) − (−1)αI(1)(p at+ A(tat), tat) R+ O(R2), (35) where Iat(p, t)= −iE(t) 2 ∂ ¯φs ∂p(p), (36) I(1)(p, t)=E(t) 4 φ¯s(p)+ iω2A(t) 4p ∂ ¯φs ∂p(p) − iE2(t) 4p ∂2φ¯s ∂p2 (p). (37)
We checked numerically that the molecular prefactor Cαβ(tαβ, tαβ ) is actually very close to the atomic one. We will thus, as a first approximation, consider it to be equal to C(tat, tat) (14). The exact effect of this approximation will be
discussed in Sec.IV B.
Finally, after regrouping all the terms, the first-order ap-proximation of the total dipole (25) can be refolded into
D() = C(tat, tat)drec(pat+ A(tat), tat)
× dion(pat+ A(tat), tat)e−iSat. (38)
We have thus retrieved an expression that is similar to the atomic dipole given by Eq. (7). It is evaluated at atomic stationary points (pat, tat, tat) and involves the atomic action
Sat. However, it involves modified transition dipole matrix
elements dionand drecexpressed as
drec(p, t )= 2R(p) cos pR 2 +2i R(1)(p, t )R sin pR 2 , (39) dion(p, t)= 2Iat(p, t) cos pR 2 + 2i I(1)(p, t)R sin pR 2 . (40)
TABLE II. Numerical values of the time deviations tαβ− tat and tαβ − tat for the same harmonics and the same generation conditions as
in TableI, compared with our development (26), to either the first or the second order. We present the numerical values for the short (S) and the long (L) trajectories. We do not present the imaginary part of tαβ− tat(<3× 10−3 a.u.), nor the real part of tαβ − tat (< 5× 10−3 a.u.).
The deviations from the atomic case are small compared to the imaginary part of the ionization times (7.83–15.8 a.u.) and the real part of the recollision times (32.9–107 a.u.), respectively.
Re(tαβ− tat) Im(tαβ − tat)
Numerical Second order First order Numerical Second order First order
αβ H S L S L S L S L 11 21 − 0.643 − 0.601 − 0.639 − 0.599 − 0.624 − 0.669 − 0.681 − 0.668 − 0.679 − 0.657 31 − 0.463 − 0.441 − 0.463 − 0.441 − 0.452 − 0.670 − 0.681 − 0.670 − 0.679 − 0.657 41 − 0.381 − 0.365 − 0.381 − 0.365 − 0.372 − 0.672 − 0.681 − 0.672 − 0.679 − 0.657 51 − 0.331 − 0.317 − 0.331 − 0.317 − 0.324 − 0.674 − 0.681 − 0.673 − 0.679 − 0.657 61 − 0.298 − 0.284 − 0.298 − 0.284 − 0.290 − 0.676 − 0.681 − 0.675 − 0.679 − 0.657 71 − 0.277 − 0.255 − 0.277 − 0.255 − 0.265 − 0.678 − 0.680 − 0.677 − 0.680 − 0.657 12 21 0.605 0.650 0.608 0.648 0.624 − 0.668 − 0.681 − 0.668 − 0.679 − 0.657 31 0.441 0.463 0.441 0.463 0.452 − 0.670 − 0.681 − 0.670 − 0.679 − 0.657 41 0.364 0.380 0.363 0.379 0.372 − 0.672 − 0.681 − 0.672 − 0.679 − 0.657 51 0.316 0.330 0.316 0.330 0.324 − 0.674 − 0.681 − 0.673 − 0.679 − 0.657 61 0.282 0.297 0.282 0.297 0.290 − 0.676 − 0.681 − 0.675 − 0.679 − 0.657 71 0.254 0.276 0.253 0.276 0.265 − 0.678 − 0.680 − 0.677 − 0.680 − 0.657 21 21 − 0.642 − 0.601 − 0.639 − 0.599 − 0.624 0.647 0.637 0.646 0.636 0.657 31 − 0.463 − 0.441 − 0.463 − 0.441 − 0.452 0.646 0.637 0.645 0.636 0.657 41 − 0.381 − 0.365 − 0.381 − 0.365 − 0.372 0.644 0.637 0.643 0.636 0.657 51 − 0.331 − 0.317 − 0.331 − 0.317 − 0.324 0.642 0.637 0.641 0.636 0.657 61 − 0.298 − 0.284 − 0.298 − 0.284 − 0.290 0.641 0.637 0.640 0.636 0.657 71 − 0.277 − 0.255 − 0.277 − 0.255 − 0.265 0.639 0.638 0.638 0.636 0.657 22 21 0.605 0.650 0.608 0.648 0.624 0.647 0.637 0.646 0.636 0.657 31 0.441 0.463 0.441 0.463 0.452 0.646 0.637 0.645 0.636 0.657 41 0.364 0.380 0.363 0.379 0.372 0.644 0.637 0.643 0.636 0.657 51 0.316 0.330 0.316 0.330 0.324 0.642 0.637 0.641 0.636 0.657 61 0.283 0.297 0.282 0.297 0.290 0.641 0.637 0.640 0.636 0.657 71 0.254 0.276 0.253 0.276 0.265 0.639 0.638 0.638 0.636 0.657
These expressions are reminiscent of the ones of [56,57], where field dressed states were used a priori to define the dipole.
III. NUMERICAL SIMULATIONS A. Ab initio computations
As presented in Sec.II A, we first investigate the signature of two-center interference in both the short and long trajectory contributions to the HHG spectrum with our TDSE computa-tions. The results are presented in Figs.2 and3. We report the intensity and phase of the short and long trajectory con-tributions to HHG spectra for several internuclear distances between 1.4 and 1.7 a.u., generated at a laser peak intensity of 2.65× 1014 W cm−2. The calibration with a reference
atom, as explained in Sec.II, implies to divide the harmonic intensity by that of the atom. Consistently, the reported phase is the phase difference between the molecular and the atomic computations, which removes the quadratic contribution and allows the observation of the two-center interference effects.
In the intensity of each spectrum we observe a minimum, which is shallow in the case of the short trajectories [Fig.2(a)] and much more pronounced for the long trajectories, espe-cially for particular values of R [Fig.3(a)]. This minimum is the signature of the destructive interference induced by the
two centers of the molecule in which we are interested. Its position has long been ascribed to the zeros of the recombi-nation dipole matrix element drec given by Eq. (16), which
relies on plane waves (PW) to describe the continuum and on the LCAO approximation for the ground state [65]. Within these approximations, the zeros of drecoccur at particular
mo-mentum values pk= (2k + 1)π/R, k ∈ Z. We focus on the first value p0as it is the only one contained in our harmonic
spectra generated by a relatively short-wavelength laser like Ti:sapphire. The corresponding harmonic frequency is
(PW) 0 = Ip+ p02 2 = Ip+ π2 2R2. (41)
For the illustrative case R= 1.57 a.u., this leads to a min-imum at harmonic 45, while our TDSE simulations give a minimum at “harmonic” 38.5.
A way to overcome this discrepancy is to claim that effec-tive PW representing the returning electron must be energy-shifted by Ip[65], leading to the approximate expression
¯ 0
π2
2R2. (42)
Still, this correction is not sufficient as it predicts a minimum at harmonic 35.
Therefore, we used numerical simulations to better un-derstand why the analytical formula, either with or
with-10−2 10−1 100 (a) -0.9π -0.6π -0.3π 0.0π 17 21 25 29 33 37 41 45 49 53 57 61 65 (b) In tensit y (arb. units) 1.4 1.45 1.5 1.55 1.6 1.65 1.7 R (a.u.) Phase (rad.) Harmonic order Ω/ω
FIG. 2. Ab initio computations of the contribution of the short trajectories to high-order harmonic spectra for a two-cycle laser pulse of 2.65× 1014W cm−2peak intensity for a diatomic molecule with
different internuclear distances R from 1.4 to 1.7 a.u. We report the harmonic intensity (a) and phase (b) calibrated to an atomic reference.
out the Ip shift [Eqs. (42) and (41), respectively] does not
reproduce the position of the minimum. We computed nu-merically the eigenstates of the field-free system by solving the time-independent Schrödinger equation. The ground state was computed by diagonalizing the field-free Hamiltonian, while the continuum states were computed with a fourth-order Runge-Kutta algorithm and normalized with the Strömgren procedure [80]. We then used these numerically exact states to compute the “exact” recombination dipole, and to extract its zero for different values of R. This is shown with a solid green line in Fig. 4. As predicted by the three-step model, the zeros of the exact recombination dipole perfectly match the position of the minima observed in the HHG spectra, as extracted from the long trajectory contributions. Note that, for small R 1.6 a.u., the mismatch is related to the difficulty of finding the minimum in the TDSE HHG spectrum, because, as can be seen in Fig.3, it is located too close to the cutoff. However, if we compute the recombination dipole using the “exact” continuum states and the LCAO ground state (dashed blue line), the position of the minimum is clearly underes-timated, by at least 10 harmonic orders. On the contrary, if we compute the recombination dipole with the “exact” ground state and PW for the continuum (dotted pink and purple lines), the position of the minimum is now greatly overestimated, whether we use the Ip shift correction or not. Finally, when
we perform both the LCAO and the PW approximations (dash-dotted orange and yellow lines), we see that the errors from both approximations compensate each other, providing a relatively good, but somehow accidental, agreement with the TDSE results. We note that this error compensation leads to a better agreement with the Ipshift correction. We thus want to
10−4 10−3 10−2 10−1 100 (a) -0.8π -0.4π 0.0π 0.4π 0.8π 1.2π 17 21 25 29 33 37 41 45 49 53 57 61 65 (b) In tensit y (arb. units) 1.4 1.45 1.5 1.55 1.6 1.65 1.7 R (a.u.) Phase (rad.) Harmonic order Ω/ω FIG. 3. Same as in Fig.2for the long trajectories.
draw attention to the fact that one always has to remain careful when considering a model with several approximations. These error compensations can actually happen more often than we think [81].
We now turn to the shape of the minimum and of the cor-responding phase jumps. As already observed in [64,65,69], the phase jumps are smoothed, covering several harmonics, with a magnitude lower than π rad [Figs. 2(b) and 3(b)]. The interference condition exhibits more surprising features. While the phase jump is smoothed and negative for all values of R for short trajectories [Fig.2(b)], the situation is completely different for the long trajectories. It is smoothed and positive for small R, then it varies steeply at a critical internuclear distance Rc= 1.57 a.u., beyond which the be-havior reverses into a smooth negative jump as for the short trajectories [Fig.3(b)]. Moreover, the depth of the minimum in the amplitude is directly related to the shape of the phase jump: the steeper the jump, the deeper the minimum.
In addition, we performed a time-frequency analysis of the harmonic dipole using Gabor transforms as in [23,56,57] to retrieve, for different values of R, the emission times of the harmonics for which the destructive interference occurs. Figure5 displays the Gabor transform of the dipole for the particular case of R= Rc. We found that the recollision time for the long trajectory associated with the interference frequency corresponds to an almost zero electric field (Erec
9.5× 10−3a.u.). Moreover, greater (lower) values of R lead to recombination in the presence of a negative (positive) electric field for the long trajectories, while the field at recombination is always positive for the short trajectories.
10 20 30 40 50 60 70 80 90 100 1 1.5 2 2.5 3 Ω0 /ω R (a.u.) No approximation LCAO PWA PWA (Ipshift) LCAO + PWA LCAO + PWA (Ipshift)
TDSE
FIG. 4. Harmonic order of the minimum as a function of R. Black dots: minimum in the harmonic spectrum extracted from TDSE calculations, lines: zero of the recombination dipole matrix element computed at different levels of approximation.
To further examine the relation between the value of the electric field at the recombination and the shape of the phase jump, we tuned the laser peak intensity IL from 2.5 to
4.5× 1014 W cm−2at fixed R. We chose R= R
eq= 1.4 a.u.
The corresponding trajectory-resolved phases are shown in Fig.6. As expected, the position of the destructive interference
10 20 30 40 −0.1 −0.05 0 0.05 0 0.5 1 1.5 2 Harmonic order Ω /ω 1x10-6 1x10-4 1x10-2 1 Harmonic intensity (arb. units)
(a) E (t ) (a.u.) t/T (b)
FIG. 5. Gabor transform of the ab initio dipole for the critical internuclear distance Rc= 1.57 a.u. (a) and instantaneous value
of the electric (b). The peak intensity is 2.65× 1014 W cm−2. We
marked with dashed lines the time associated with the minimum observed in the long trajectories.
-0.9π -0.6π -0.3π 0.0π (a) -0.8π -0.4π 0.0π 0.4π 0.8π 1.2π 17 21 25 29 33 37 41 45 49 53 57 61 65 69 (b) Phase (rad.) 2.5 3 3.5 4 4.5 Laser intensity (1014Wcm−2) Phase (rad.) Harmonic order Ω/ω
FIG. 6. Ab initio computations of the harmonic phase of a model H2 relative to the atomic reference for a two-cycle laser pulse with
peak intensities in the range (2.5–4.5)× 1014 Wcm−2 for short (a)
and long trajectories (b). The dashed line corresponds to intensity
Ic= 3.74 × 1014W cm−2.
does not change with IL and we observe a behavior for the
phase jumps that is similar to the one when varying R. It is smoothed and negative for all intensities at the short trajecto-ries, while an equivalent smoothing and inversion behavior is present for the long trajectories. We found a critical intensity Ic= 3.74 × 1014W cm−2at which the inversion occurs with a discontinuous ±π rad phase jump [see the black-dashed curve in Fig.6(b)].
Using time-frequency analysis once again, this change of behavior is directly related to the change of sign of the instantaneous electric field at recombination time, which is almost zero at 0 for the intensity Ic. We find that, in this case, the corresponding electric field amounts to Erec= 7.4 × 10−3 a.u. This corresponds to ≡ Irec=
1.92× 1012Wcm−2, which is∼ 0.5% of Ic.
B. Molecular SFA
To get more physical insights, the same study has been carried out within the molecular SFA derived in Sec. II C, following [71]. We report in Fig. 7 the phase of the short and long trajectories as a function of the laser intensity. Again this phase is relative to the phase of atomic SFA presented in Sec. II B, following [60]. The separation of the short and long trajectory contributions in SFA is straightforward, being inherent to the search of saddle-point solutions. Consequently, no residual oscillation is encountered, in contrast to the TDSE results. Nevertheless, the SFA model prevents the computa-tion of harmonics above the cutoff, which limits the observa-tion of the entire phase jumps for the lowest intensities.
We observe that the behavior of the phase is very similar to that of the TDSE simulations. The major difference is the
-0.9π -0.6π -0.3π 0.0π (a) -0.8π -0.4π 0.0π 0.4π 0.8π 1.2π 37 41 45 49 53 57 61 65 69 (b) Phase (rad.) 3 3.5 4 4.5 5 Laser intensity (1014Wcm−2) Phase (rad.) Harmonic order Ω/ω
FIG. 7. Molecular SFA computations of the harmonic phase of a model H2normalized by the atomic SFA for a two-cycle laser pulse
with peak intensities in the range (3–5)× 1014Wcm−2for short (a)
and long trajectories (b). The dashed line corresponds to the intensity
Ic= 3.704 × 1014W cm−2.
position of the destructive interference due to the LCAO and PW approximations in SFA, as mentioned before. Hence the destructive interference is observed exactly at the frequency given by Eq. (41). Furthermore, the intensity for which the phase is discontinuous and jumps by ±π rad is shifted to Ic= 3.704 × 1014W cm−2.
The SFA computations allowed us to obtain the precise value of the electric field at recombination time for the long trajectories and for the inversion intensity Ic. It is found to be Erec= 0.0034 a.u., which is comparable to the one
observed in the ab initio computations. The discrepancies are attributed to the difference between ab initio and SFA trajectory times [82].
Finally, we have performed simulations at the laser inten-sity for which the electric field is exactly zero at recombination time—Iz= 3.641 × 1014 W cm−2, which is slightly lower than Ic, with a difference of only 6× 1012 W cm−2, i.e., <2%—and we found that the jump is not discontinuous and lower than−π rad.
C. Quantum path interferences
We have seen in the previous two sections that the HHG phase behavior depends strongly on the type of trajectory (short versus long) and on the value of the electric field at recombination time, thus entangling the structural information contained in the two-center interferences and the electron dynamics.
In practice, the influence of the electron dynamics on a structural interference could be observed by following the phase difference between the short or long trajectory contri-butions when varying the laser intensity, e.g., by recording
-4 -2 0 2 Phase Difference ∆φ (rad) 19 13 25 31 37 43 49 55 61 67 73 79 1 2 3 4 5 6 7 Laser Intensity (10 Wcm )14 -2 Harmonic Order 19 13 25 31 37 43 49 55 61 67 73 79 (a) (b) Short Long
Harmonic intensities (arb. units, log scale)
FIG. 8. (a) QPI for the model molecule with R= 1.4 a.u. (blue lines) and the atomic reference (black lines) from TDSE computa-tions. The curves corresponding to the different harmonic orders are shifted vertically for clarity; (b) phase differenceϕ between the molecular and the atomic IL→ α Fourier transform components of
the QPI, associated with the short (red lines) and long (green lines) trajectory contributions.
the quantum path interferences (QPI) in the total harmonic dipole [83,84]. We performed such an analysis in our TDSE simulations, and we show the results in Fig.8(a)for both the molecule and the reference atom. For each harmonic, we ob-serve oscillations corresponding to the interferences between the short and long trajectories. They mainly come from the phases accumulated by the electron during its excursion in the continuum, i.e., during the second step in HHG, which naturally depends on the trajectory duration. However, for molecules, one also has to take into account the phase of drec,
which, as seen above, depends significantly on the continuum electron dynamics and is different for each contribution. In fact, as we can see in Fig. 6, for harmonics far from the interference (H49), the recombination dipole phase difference for the two trajectory contributions is either≈0 or ≈2π rad, leading to oscillations in phase with the reference atom. For harmonics around H49, the phase difference is≈ π rad and the QPI exhibit oscillations out of phase with the reference atom. This clear signature makes the QPI a method of choice to observe a pure structural interference modified by electron dynamics.
We extracted these phases by performing an IL → α
Fourier transform of the QPI, as described in [85]. We present in Fig. 8(b) the phase difference ϕ between the model molecule and the reference atom. We observe phase jumps that are an average of the different phase jumps over the laser intensity, and that clearly reproduce the
ex-pected behavior, i.e., smooth π jumps occurring around H49, in opposite directions for the long and short trajectories, respectively.
Below, we shall show how the Taylor expansions of the molecular SFA times, action, and dipole presented in Sec. II D give a comprehensive understanding of the vari-ations of the HHG dipole phase jump with respect to R and IL.
IV. INTERPRETATION
In this section, we examine the dipole D() given by Eq. (38), investigating each of its terms (see Sec.II D 2). We focus in particular on the role of the modified recombination dipole matrix element drecof Eq. (39).
A. Recombination dipole matrix element
1. Explanation of the phase jumps
We already explained in Sec. III that the zero of drec
[Eq. (16)] encountered at p0, see Eq. (41), encodes the
posi-tion of the destructive interference. For symmetry reasons (the ground-state wave function is even), this recombination dipole matrix element is a real number in the PW approximation. Thus, it is constrained to the real axis and its evolution around p0from a positive to a negative value implies a discontinuity
of its phase, with a jump of exactly−π rad.
However, the modified recombination dipole matrix ele-ment drecexpressed in Eq. (39) evolves in a totally different manner around p0. As it contains a nonzero imaginary part, it
is no longer constrained to the real axis and describes a path in the complex plane when varying the harmonic order. It turns out that this imaginary part is proportional to the instantaneous electric field at the recombination time; see Eqs. (33) and (39). Figure9(a)displays the values of drec in the complex plane for the short trajectories, for different laser intensities, when scanning the harmonic order. The path in the complex plane, followed at a fixed intensity, crosses the Re(drec)= 0 line for
a nonzero imaginary part, implying a smooth variation of the phase. In addition, we observe that the higher the electric field at recombination, the further the path is from the origin, and consequently the smoother is the phase jump. Furthermore, as the electric field at the recombination is always positive for the short trajectories, as pointed out in Sec.III, we observe a systematic negative phase jump as shown in Fig. 9(b). This behavior is fully imprinted on the phase of the full HHG dipole as seen in Fig. 7(a). Note that for the lowest intensities, the phase jump is not complete due to the low cutoff value.
The situation is completely different for the long trajecto-ries for which the recombination times are distributed across two laser half-cycles: the electric field is first positive, crosses zero, and is then negative when decreasing the harmonic order. As we can see in Fig.10(a)for the particular intensity denoted Iz (black dashed line), the path followed by drec
crosses the origin. This corresponds to a zero electric field at recombination time and thus a zero imaginary part, when the real part also cancels. Therefore, we retrieve the ideal case for which the phase is discontinuous and jumps to±π
-4 -3 -2 -1 0 -2 0 2 4 6 8 10 (a) H -0.9π -0.6π -0.3π 0.0π 37 41 45 49 53 57 61 65 69 (b) Im (10 − 3 a.u.) Re (10−3 a.u.) 3 3.5 4 4.5 5 Laser intensity (1014Wcm−2) Phase (rad.) Harmonic order Ω/ω
FIG. 9. Modified recombination dipole matrix element drec
[Eq. (39)] in the complex plane for the short trajectories (a). The different paths correspond to different laser intensities between 3 and 5× 1014 W cm−2 and for the specific intensity of
Iz= 3.641 × 1014W cm−2(black dashed line). Each path shows the
evolution of drecwith the harmonic order H (see black arrow). Note
that the position of the destructive interference corresponds approxi-mately to the intersection between the curves and the imaginary axis (i.e., zero real part). The phase of drecis reported in (b).
rad [see Fig. 10(b)]. For the other intensities, the laser field is not zero and the phase evolves slowly, varying from zero to −π (π) rad for a positive (negative) electric field at the recombination.
Thus, we fully understand the behavior of the recombina-tion dipole matrix element drec, which mainly drives the HHG
phase evolution along the two-center destructive interference. Its deviations from the ideal case are governed by the con-tinuum dynamics of the electron during the second step of HHG. This leads to a phase jump whose shape depends on the instantaneous value of the electric field at recombination time. Since the electrons following short or long trajectories recombine with the molecule within different time windows, the corresponding phase jumps behave completely differently. As we explained, three different behaviors are notably ob-served for the long trajectories. We analyze below quantita-tively the conditions for which we can observe these three situations.
-0.5 0 0.5 1 1.5 2 -2 0 2 4 6 8 10 (a) H -0.9π -0.6π -0.3π 0.0π 0.3π 0.6π 0.9π 1.2π 37 41 45 49 53 57 61 65 69 (b) Im (10 − 3 a.u.) Re (10−3 a.u.) 3 3.5 4 4.5 5 Laser intensity (1014Wcm−2) Phase (rad.) Harmonic order Ω/ω
FIG. 10. Same as in Fig. 9 but for the long trajectories. The particular path that crosses the origin corresponds to an intensity of
Iz= 3.641 × 1014W cm−2(black dashed line).
2. Inversion condition
In this section, we quantify the conditions to observe the inversion in the phase jump for the long trajectories in our simulations either as a function of internuclear distance R (see Fig.3in Sec.III A) or as a function of the laser intensity IL
(see Fig.6in Sec.III Aor Fig.7in Sec.III B).
For the long trajectory contributions, the electric field changes sign at a single harmonic frequency denoted 0.
Hence, harmonics below the cutoff and beyond (below) 0
are emitted in the presence of a positive (negative) electric field.
The value of 0 can be obtained with atomic classical
calculations [58]. We thus solved the Newton equation for an electron freed at ionization time t and brought back to its initial position x = 0 at recombination time t under the influence of the laser field only. The recollision time for which the laser field is zero is tc= 3π/2ω [see Eq. (5)], leading to
the corresponding ionization time:
tc 1 3π 2 − 2 3π ω. (43)
Thus, the corresponding critical harmonic frequency is
0= Ip+
EL2 2ω2[sin(ωt
c)+ 1]2≈ Ip+ 2.98Up, (44)
where Up= EL2/4ω2= IL/4ω2is the ponderomotive energy.
Therefore, we have two knobs to tune the relative values of 0(PW), Eq. (41), and 0, Eq. (44), i.e., the internuclear
distance R for the former and the laser intensity for the latter. By varying these knobs, we can put our system in the three cases that we previously discussed: (i)0(PW)<0, the phase jump for the long trajectories is smoothed and has an opposite direction from that of the short trajectories; (ii)(PW)0 = 0,
the phase jump for the long trajectories is discontinuous and is equal to±π rad; (iii) 0(PW)>0, the phase jump for the
long trajectories is smoothed and has the same direction as that of the short trajectories.
Furthermore, we found that the equality of(PW)0 and0
can be obtained for an infinity of couples (IL, R),
approxi-mately verifying the relation R 2.58√ω
IL
. (45)
Finally, we would like to note that, except in our ab initio computations, the entire phase jump is not observed because 0is close to the cutoff frequency, i.e., Ip+ 3.17Up [86], as we can see in Figs.10(a)or10(b). Unfortunately, one cannot expand the region between these two frequencies by changing the laser intensity since both depend linearly on it.
The SFA-based analysis of drec thus allows us to relate the discontinuous phase to the recombination in the pres-ence of a zero electric field. However, in Sec. III, both for TDSE and molecular SFA computations, we observed that the sharpest phase jump is obtained when the electric field at recombination time is not exactly zero. In the following section, we derive the first-order expansion of the prefactor Cαβand explain the origin of this shift.
B. Prefactor correction
When we obtained the first-order expression of the SFA molecular dipole in Sec.II D 2, we neglected the first order of the molecular prefactor Cαβ. This first approach allowed us to get a factorized expression for the molecular dipole (38) that led us to important physical insights into the inversion of the π jump observed in the HHG phase associated with the long trajectories. However, as mentioned above, discrepancies remain as to the exact values of the electric field for which we observe the phase-jump inversion. To assess the origin of this difference and to go deeper into the analysis, we complete here the analytic derivation of the expression of Cαβup to the
first order as Cαβ(tαβ, tαβ )= Cat(tat, tat) 1+R 2 (−1)α E(tat) 2[pat+ A(tat)]2 + (−1)βE(tat) 2[pat+ A(tat)]2 + (−1)α (tat− tat)[pat+ A(tat)] − (−1)β (tat− tat)[pat+ A(tat)] . (46)
For details, see AppendixB.
We then need to sum over the four trajectories (i.e., over αand β) to obtain the full first-order expression of the total dipole (25). The latter can again be factorized in the same form as in the atomic case (7):
D() = C(tat, tat)drec(pat+ A(tat), tat, tat)
× dion(pat+ A(tat), tat, tat)e−iSat+ O(R 2) , (47)
where the modified recombination drec and ionization dion dipoles now include contributions from the prefactor expan-sion. Their expressions read
drec(p, t, t)= 2R(p) cos pR 2 + 2i Q(p, t, t)R sinpR 2 , (48) with Q(p, t, t)= −E(t ) 4 ∂ ¯φs ∂p (p)− E(t ) 8p φ¯s(p)− ¯ φs(p) 4(t− t) (49) and dion(p, t, t)= 2Iat(p, t) cos pR 2 + 2i K(p, t, t)R sinpR 2 , (50) with K(p, t, t)= E(t) 4 φ¯s(p)− i E2(t) 4p ∂2φ¯ s ∂p2 (p) + i∂ ¯φs ∂p (p) 3 2(t− t)p − E(t) 4Ip +ω2A(t) p . (51) We thus recover the same kind of expressions as before, with additional terms in the modified recombination drecand
ioniza-tion diondipoles that account for the influence of the prefactor.
In fact, these expressions are reminiscent of Eqs. (33) and (36), whereR(1) andI(1) defined in Eqs. (39) and (40) are
now replaced byK and Q, respectively. Note that these extra terms correspond to the ζ parameter introduced in Eq. (4) of Ref. [72]. When looking at the expression of drec, we remark
that, because of these new terms, the imaginary part of drec,
which is related to the real part ofQ in Eq. (49), is no longer proportional to the electric field at recombination time. It thus
1.25 1.3 1.35 1.4 1.45 1.5 1.55 3.2 3.4 3.6 3.8 4.0 4.2 4.4 R (a.u.) Laser intensity (1014Wcm−2) 10−8 10−7 10−6 10−5 10−4
Harmonic intensity (arb. units)
FIG. 11. Value of the minimum of the intensity of the long trajectory contribution in the HHG spectrum as a function of the laser intensity and nuclear distance R. The colormap corresponds to the value of the minimum of the molecular SFA dipole Eq. (25). The lines corresponds to the couples (R, IL) for which the modified
recombination dipole crosses zero in the complex plane: solid red line drec, Eq. (39); dotted white line drec, Eq. (48).
explains why the sharp phase jump occurs for a small nonzero electric field at recombination time.
To better understand the effect of these new terms in the modified recombination dipole, we compare both drecand drec to the full SFA results in Fig.11. The color map indicates the value of the minimum in the full SFA dipole Eq. (25). The darkest pixels correspond to the couples (R, IL) for which the
minimum is the deepest and the phase jump is the sharpest. The lines show the couples (R, IL) for which the modified
recombination dipole crosses the origin of the complex plane: the solid red line is for drec(without the prefactor contribution)
and the dotted white line is for drec(with the prefactor
contri-bution). It is striking that the prefactor contribution allows us to completely recover the correct behavior. Indeed, the zeros of drec perfectly match the deep minima in the SFA spectra. As a consequence, the study of the modified recombination dipole drecis sufficient to fully recover the position and the
shape of both the structural minimum and the phase jump expected in diatomic molecular HHG.
V. CONCLUSION
In this paper, we have revisited the phenomenon of two-center interferences in molecular HHG, with a strong focus on its spectral phase. Using both TDSE and molecular SFA computations, which display similar results, we have shown that the phase for the short and long trajectory contributions behave in totally different ways. In particular, the destructive interference phase jump for the former is always smooth and negative, while for the latter the features of the jump strongly depend on the internuclear distance and on the laser intensity.
